Suppose I have a fibre bundle $E\to B$ with compact fibre. Furthermore, $B$ is open in a larger, compact space, e. g. $B\subseteq B'$. I want to get a map $E'\to B'$ (not a bundle any more!) with
- $E'|_B = E$
- For $x\in B'\setminus B$ each fibre is only one point.
- $E'$ is a compact space.
- Each section $\omega:B\to E$ can be extended to $\omega':B'\to E'$.
The idea is to »collapse« the whole fibre when reaching the boundary of $B$.
For example, if $E:=[0,1]\times (0,1)$, $B:=(0,1)\subseteq [0,1]:=B'$ and the bundle $E\to B$ is the projection, this should be extended on $B':=[0,1]$ to $E'=\Sigma [0,1]$.
Is there a precise way to give this topological construction explicitely?